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A quadrilateral has vertices [tex][tex]$A(11,-7), B(9,-4), C(11,-1)$[/tex][/tex], and [tex][tex]$D(13,-4)$[/tex][/tex].

Quadrilateral [tex][tex]$ABCD$[/tex][/tex] is a [tex][tex]$\square$[/tex][/tex].

If point [tex][tex]$C^{\prime}(11,1)$[/tex][/tex] is given, quadrilateral [tex][tex]$ABCD^{\prime}$[/tex][/tex] would be a [tex][tex]$\square$[/tex][/tex].



Answer :

Let’s analyze the problem in a step-by-step manner to understand the types of quadrilaterals formed by the given vertices.

First, let's consider quadrilateral [tex]\(ABCD\)[/tex] with vertices [tex]\(A(11, -7)\)[/tex], [tex]\(B(9, -4)\)[/tex], [tex]\(C(11, -1)\)[/tex], and [tex]\(D(13, -4)\)[/tex].

To determine the type of quadrilateral [tex]\(ABCD\)[/tex]:
1. Check for any unique combination of lengths or parallel sides which might suggest a specific type of quadrilateral such as an isosceles trapezoid, parallelogram, or rectangle etc.

However, upon analyzing the coordinates:
- Vertices [tex]\(A\)[/tex] and [tex]\(C\)[/tex] share the same x-coordinate (11), but have different y-coordinates (-7 and -1 respectively).
- Vertices [tex]\(B\)[/tex] and [tex]\(D\)[/tex] share the same y-coordinate (-4).
- The other coordinates do not provide enough symmetry or parallel properties that typically define a specific type.

Given this, quadrilateral [tex]\(ABCD\)[/tex] cannot be easily identified as a standard geometric figure just by given coordinates.

Hence, [tex]\( \boxed{\text{unspecified}} \)[/tex] is the correct categorization for the quadrilateral [tex]\(ABCD\)[/tex].

Next, let’s consider quadrilateral [tex]\(ABC'D\)[/tex] where point [tex]\(C'\)[/tex] is at [tex]\((11, 1)\)[/tex]:
Here [tex]\( \overline{BC'} \)[/tex] is vertical and since [tex]\( \overline{AD} \)[/tex] is the only base which could be shared by a trapezoid observing y-coordinates doesn't lead to an identifiable pattern as definitive type like rectangle or parallelogram.

Hence, [tex]\( \boxed{\text{unspecified}} \)[/tex] is again correct categorization for the quadrilateral [tex]\(ABC'D\)[/tex] as well.

Thus, filling in the blanks, we can conclude:
Quadrilateral [tex]\(ABCD\)[/tex] is an [tex]\(\text{unspecified}\)[/tex], and with point [tex]\(C^{\prime}(11,1)\)[/tex], quadrilateral [tex]\(ABC^{\prime}D\)[/tex] would also be [tex]\(\text{unspecified}\)[/tex].

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